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A map’s straight-line distance measures the direct separation between two points. A journey length measures the path actually followed. A correct scale conversion can therefore give the wrong answer to a route question if the student measures the direct line instead of the stated route.
For G1 Mathematics, decide which distance is requested before using the map scale. Trace the permitted path, find its map length, and then convert using consistent units. The scale tells you how map length relates to represented length; it does not choose the path for you.
If your child uses the right scale but gets a shorter answer, the calculation may already be sound. Check what they measured. The missing decision may be the difference between a direct separation and a route, rather than a weakness in multiplication or unit conversion.
Contents
Sections 1–5
Sections 6–10
Sections 11–15
Sections 16–20
Sections 21–25
Sections 26–29
Imagine a flat scale drawing with a route that goes three centimetres east and then four centimetres north. The route length on the map is seven centimetres. The straight segment joining its endpoints is five centimetres, by Pythagoras’ theorem.
At a scale of 1:25,000, one centimetre represents 250 metres. The route therefore represents 1,750 metres, while the direct separation represents 1,250 metres. Both conversions are correct, but they describe different lengths.
The five-centimetre segment is not automatically an available route. It might cross an area the invented problem does not permit the traveller to enter. Only the stated path determines the journey being calculated.
All maps and journeys here are invented classroom models. Unless a section says otherwise, the drawing uses a uniform scale on a flat plane, with no elevation or real-world access complication beyond the stated paths. The purpose is to separate measurement choice from scale conversion clearly.
The SEAB G1 Mathematics syllabus for 2027 includes map scales, ratios, units and practical applications. A map problem combines these ideas with reading: the learner must identify the length that the question actually asks them to convert.
This guide concerns simple scale drawings and invented route networks. It does not claim that a classroom drawing predicts an actual walking route, road closure or current travel time. Real journeys require current route information in addition to any map measurement.
The broader G1 Mathematics learning guide places practical numeracy within the subject. Here the focused decision is whether the distance refers to direct separation or travel along a specified path.
G1 is a subject level, not a label for a child’s potential. A student can understand ratio conversion yet still need help deciding which line to measure. Teaching that choice explicitly allows their existing arithmetic knowledge to be used more effectively.
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Direct separation is the length of the straight segment joining the start and finish in the flat drawing. Route length is the accumulated length of the path followed between them. Their endpoints can match while their lengths differ.
In the opening example, both measurements begin at the same starting point and end at the same destination. The difference comes from the path between those points. The route turns a corner, while the direct segment cuts across it.
Use clear labels such as “route AB via C” and “direct AB.” A vague label like “distance AB” can conceal which path was used when the diagram shows several possibilities. The words via, along and following the path often carry essential information.
Before writing a formula, ask the learner to trace the requested distance with a finger or describe it in words. This is a low-effort way to catch a model error before it becomes a full page of correct arithmetic applied to the wrong length.
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A scale of 1:25,000 means one length unit on the map corresponds to 25,000 of the same length unit in the represented model. One centimetre therefore represents 25,000 centimetres, which is 250 metres or 0.25 kilometres.
The ratio does not mean one centimetre represents 25,000 metres. That would mix units before the comparison is interpreted. Convert the represented centimetres into metres or kilometres only after applying the scale correctly, or establish an equivalent per-centimetre conversion first.
For five map centimetres, the represented length is 125,000 centimetres, or 1,250 metres. For seven map centimetres, it is 175,000 centimetres, or 1,750 metres. The common scale factor preserves the distinction between those two map lengths.
The quantity-reading guide supports this first check. A number’s role and unit determine the operation. Keep the ratio meaning visible rather than memorising an unexplained instruction to multiply by a large printed number.
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A reliable solution has two stages. First find the map length of the requested path. Then convert that length through the scale. If the first stage uses the wrong path, a flawless second stage will still describe the wrong quantity.
For the opening route, the first stage is three plus four equals seven map centimetres. The second is seven multiplied by 250 equals 1,750 metres. For the direct segment, the first stage is the Pythagorean calculation giving five centimetres, followed by five multiplied by 250.
Do not choose the five-centimetre answer simply because it uses a more advanced-looking theorem. Pythagoras answers a particular geometric question. It does not override the route specified in the wording.
Writing stage labels can help: “Map route length” followed by “Represented route length.” This makes it easier for a teacher, parent or student to identify whether an incorrect result came from measuring the wrong object or from converting the right object incorrectly. The repair should target the first unsupported step.
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One table can hold both legitimate measurements without pretending that they are interchangeable. The description column is as important as the numerical columns.
| Measurement | Map length | Represented length at 1:25,000 |
|---|---|---|
| Three-centimetre plus four-centimetre route | 7 cm | 1,750 m |
| Direct segment between the same endpoints | 5 cm | 1,250 m |
| Route excess over direct separation | 2 cm | 500 m |
The difference of five hundred metres is not a scale-conversion error. It is the extra path length created by following the two perpendicular legs instead of the direct segment. The scale faithfully carries that difference into the represented model.
If the question asks for the journey along the two legs, 1,750 metres is the relevant answer. If it asks for the straight-line distance, 1,250 metres is relevant. A numerical result becomes meaningful only when its label matches the requested measurement.
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For paths on a flat plane, a straight segment is no longer than any route joining the same endpoints. A route can equal it if it follows that direct segment without backtracking. A turning or detouring route can be longer.
This gives a useful error check. If a calculated route length is shorter than the correctly calculated direct separation in the same scale model, something needs review. A segment may have been omitted, a unit changed incorrectly, or the measurements may describe different endpoints.
Passing the lower-bound check does not prove that the route length is correct. Many wrong lengths can still exceed the direct distance. The bound rejects impossible comparisons under the model; the traced route and its measurements establish the actual answer.
State the flat-model condition when using this reasoning. A schematic transit diagram may not be drawn to a uniform geographic scale, and a real surface route may involve elevation. The classroom claim concerns lengths represented consistently on the stated plane, not every line drawn on every kind of map.
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Suppose an invented path has map segments of 2.4 cm, 1.6 cm and 3 cm. Its total map length is seven centimetres. At the opening scale, that again represents 1,750 metres, even though the route has three segments rather than two.
The number of turns is not itself a distance. Each segment contributes its length. A short route can contain several turns, while a long straight segment can contain none. Do not add a fixed amount merely because a corner is visible unless the problem defines such an extra distance.
Check that the segments connect in the order described. If the route is A to B to C to D, use AB, BC and CD. A nearby segment such as AC may be a shortcut or a different measurement and should not replace two required legs without permission.
A sum of named segments is useful working. It records the chosen path as well as the arithmetic, making omissions and double counting easier to spot. A single unexplained total hides the route decision from anyone reviewing the solution.
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A curved path is not measured by joining its endpoints with one straight ruler segment. That segment gives its chord-like direct separation, not the curved length. A classroom problem may supply the curved route’s length or instruct the learner how to estimate it.
One possible approximate method is to use a series of short straight segments following the curve. Another is to compare a flexible tracing with the map’s scale, if the activity permits it. The chosen method and its limitations should be stated rather than treated as exact by default.
For an invented drawing, suppose the route is measured as approximately 6.8 cm and one centimetre represents 200 metres. The estimated route length is approximately 1,360 metres. The approximation label belongs to the measured map length and carries through the conversion.
Do not invent precision beyond the drawing’s resolution. A long calculator display after multiplying an approximate length does not make the underlying route measurement exact. Follow the requested accuracy and distinguish a supplied exact classroom value from an estimate read off a printed curve.
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SECTION 10 OF 29
9. A map legend can describe a schematic rather than a scale drawing
Some diagrams are designed to show connections rather than proportional distances. An invented network might place four stations evenly apart on the page even though the represented segment lengths differ. Counting equal-looking gaps would then misread the diagram.
If the task supplies numerical lengths on the connections, use those lengths. If it explicitly states that the drawing is not to scale, the apparent centimetre lengths are not evidence of actual distance. A ruler cannot recover information the drawing does not encode proportionally.
The phrase map is therefore not enough by itself to justify measuring. Check for a valid scale statement, a scale bar or instructions authorising measurement. The representation’s job determines which operations it supports.
This connects to the guide to reading diagram marks. A diagram can support exact reasoning through given labels while its appearance remains schematic. Use the information actually supplied instead of silently promoting visual spacing into a measured fact.
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Suppose a printed scale bar spans four centimetres and represents one kilometre. One centimetre then represents 250 metres. If the bar is divided into four equal sections, each section represents 250 metres, not one kilometre.
Read whether the label refers to the complete bar or to one interval. The same error occurs when students treat every tick as one unit regardless of the marked values. Count intervals and compare their endpoints.
The scale-marks guide develops this foundation. For a map, the interval conversion must be secure before either direct or route length is converted.
If the scale bar and the route are part of the same uniformly resized image, comparing their lengths can preserve the represented ratio. A written numerical scale such as 1:25,000 depends on the drawing being displayed or printed at its intended size. The next section explains why resizing can affect those two forms differently.
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Imagine a map originally printed at 1:25,000. If its entire image is enlarged uniformly to twice its original linear size, a route that occupied five centimetres now occupies ten. The represented route itself has not become twice as long.
At the enlarged size, the correct numerical scale would be 1:12,500. Continuing to multiply the new ten-centimetre measurement by 25,000 would overstate the represented length by a factor of two.
A scale bar enlarged together with the map remains usable as a length comparison because both route and bar change by the same factor. The ratio between their displayed lengths remains unchanged. This assumes uniform resizing of the same image, not separate editing of the bar.
The mathematical lesson is to identify what changed: the paper or screen representation, not the represented geography. In a classroom task, use the supplied scale as intended. When an image has been copied or resized, avoid claiming exact represented distances from a printed ratio unless the display size or a matching scale reference is reliable.
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If a drawing is stretched horizontally by two but left unchanged vertically, horizontal and vertical lengths no longer share one common scale factor. A route with both directions cannot be converted accurately using one unchanged numerical ratio.
In the opening three-by-four example, doubling only the horizontal display changes the two leg lengths to six and four centimetres. Their displayed total becomes ten, while the original represented journey remains the same. The distorted image is no longer the original uniform scale drawing.
This does not require the learner to diagnose every possible image-editing problem. It teaches a condition behind the method: one scale factor applies only when the representation preserves the relevant length ratios consistently.
If a question explicitly provides an accurate scale drawing, accept that classroom condition. If a real screenshot has uncertain distortion, state the limitation rather than treating a precise ruler reading as decisive. Numerical precision is useful only after the representation is known to support the measurement being made.
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An invented map has scale 1:20,000. A permitted route from A to B via C consists of AC = 3.5 cm and CB = 2.5 cm on the map. Find its represented journey length. The map route length is six centimetres.
One centimetre represents 20,000 centimetres, or 200 metres. Therefore six centimetres represents 1,200 metres, equivalent to 1.2 kilometres. The route instruction via C determines which two segments were added.
If the direct segment AB is drawn as 4.8 cm, it represents 960 metres. That does not replace the 1.2-kilometre journey when the traveller must go through C. It supplies a comparison, showing a route excess of 240 metres.
Check the result backward: 1,200 metres is 120,000 centimetres. Dividing by 20,000 gives six map centimetres, matching the traced route. The direct and route values are both internally consistent; the wording tells us which one answers the question.
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If the route length is 1.2 kilometres and an invented traveller moves at a constant four kilometres per hour with no stops, the time is 1.2/4 = 0.3 hours, or eighteen minutes. The speed and no-stop conditions are additional information.
Using the 960-metre direct separation instead would give 0.24 hours, or 14.4 minutes. That would describe travel along a different length. A correct time formula cannot repair an incorrect choice of distance.
The average-speed guide develops the need to reconstruct the actual journey interval. Here the route measurement is an earlier input to that reconstruction.
Do not turn the classroom calculation into a prediction of actual arrival time without considering the real journey conditions. The mathematical example assumes a stated constant speed and no delays. Its purpose is to show how a path decision can affect a later calculation, even when every subsequent operation is performed correctly.
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A. At scale 1:10,000, convert a route of 6 cm into metres. B. At the same scale, a direct segment is 4 cm while a required route is 2 cm plus 3 cm. Find both represented lengths and their difference.
C. A flat map has perpendicular route legs of 6 cm and 8 cm. Find the route length and direct separation on the map. At scale 1:5,000, convert both to metres. D. A scale bar of 5 cm represents 2 km. What does 1 cm represent?
E. At scale 1:25,000, a represented journey is 2 km long. Find its map route length. F. A route has segments 1.2 cm, 2.8 cm and 1.5 cm at scale 1:20,000. Find its represented length in kilometres.
G. Explain why a direct separation cannot automatically be used for a journey through a required checkpoint. H. Explain what must be checked before measuring a map image that has been resized. State the measurement or representation issue, not just a numerical answer.
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A gives six hundred metres because one centimetre represents one hundred metres. B gives a direct separation of four hundred metres and a route of five hundred metres, with a one-hundred-metre excess.
For C, the map route length is fourteen centimetres. The direct separation is ten centimetres by Pythagoras. One centimetre represents fifty metres, so the route is seven hundred metres and the direct separation five hundred metres.
For D, one centimetre represents 0.4 km, or four hundred metres. For E, two kilometres is two hundred thousand centimetres; dividing by twenty-five thousand gives eight map centimetres. For F, the map total is 5.5 cm, representing 1,100 metres or 1.1 kilometres.
For G, the checkpoint changes the permitted path, so the direct segment may not describe the journey. For H, check whether the scale still matches the displayed size and whether a scale bar was resized consistently. A nonuniformly stretched image may no longer support one common length scale.
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The shortest route from A to B without restrictions need not be the shortest route from A to B through C. A required checkpoint adds a condition. Any candidate route that misses C fails the problem even if its length is smaller.
For an invented flat drawing, suppose direct AB is five units, while A to C to B totals seven. Seven is not an incorrect minimum merely because a five-unit segment exists. It may be the required route length under the checkpoint condition.
If several allowed paths through C are shown, compare only paths meeting that condition. A shorter prohibited candidate should not be included in the final choice. State the permitted-route set before deciding which total is smallest.
This connects to the shared-condition guide. A valid answer must satisfy every required condition. Correct arithmetic on an inadmissible path does not make that path an acceptable solution to the complete question.
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Suppose Route One measures 7.2 cm and Route Two measures 6.4 cm on the same uniform map. At 1:25,000, their represented lengths are 1.8 km and 1.6 km. Route Two is shorter by 0.2 km.
The common scale factor preserves the ordering. The route shorter on the map is shorter in the represented flat model. There is no need to convert both merely to decide which map length is smaller, although conversion is needed if the question asks for the represented difference.
Check that both measurements use the same map and scale. A 6.4-centimetre route on a different drawing cannot be compared directly with 7.2 centimetres here without its own conversion. Displayed lengths alone are not comparable across different scales.
Likewise, a shorter distance is not automatically a shorter time if speeds or delays differ. Keep the optimisation target clear: shortest path length, fastest journey or another criterion. The numerical comparison answers only the quantity actually calculated.
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If a traveller follows a 1.75-kilometre route outward and returns along the same route, the total distance travelled is 3.5 kilometres. Finishing at the starting point does not make the travelled distance zero.
The direct separation between the final and initial positions is zero, but that is a different measurement. This is another version of the distinction between an accumulated path and a relationship between endpoints.
If the return route differs, calculate its length separately and add it to the outward length. Do not double the outward route merely because the word return appears. Equal endpoints do not guarantee equal paths.
A useful working line is “Outward distance + return distance.” This preserves the journey structure and makes any equal-route assumption visible. The calculation can then support an average-speed question or a total-distance comparison without confusing the path travelled with the final separation from the start.
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The main examples use a flat plane. A plan view records horizontal geometry in that model. If a problem explicitly introduces a straight sloping segment with horizontal run three metres and vertical rise four metres, its actual segment length is five metres.
That is another reason to read what the map or diagram represents. Horizontal separation and sloping length are different quantities. The appropriate right-triangle calculation depends on a stated vertical dimension, not on an invented assumption about the ground.
Do not add an elevation correction to an ordinary flat-map question when none is supplied. Equally, do not ignore a stated vertical rise if the task asks for the sloping segment’s length. The model’s conditions decide which distance is relevant.
This section is a mathematical extension, not a claim that real terrain can always be represented by one straight slope. A real path may change elevation repeatedly. Classroom geometry simplifies those details explicitly; the learner should preserve the given simplification rather than confuse it with an exact survey of an actual route.
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At scale 1:25,000, one map centimetre represents 250 metres of length. A square measuring one centimetre by one centimetre on the map therefore represents a square 250 metres by 250 metres in the flat model.
Its represented area is 62,500 square metres, not 250 square metres. A length conversion factor applies in each of two perpendicular directions when converting area. The quantity being measured changes the scaling relationship.
The square-unit conversion guide develops this distinction. It is relevant when a map exercise moves from a route length to a region’s area in a later part.
Keep the answers and units separate. A kilometre measures a length, while a square kilometre measures area. The route question does not become an area question merely because a region is drawn nearby, and an area question cannot be solved by applying the single linear scale factor once.
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Suppose a route is read as approximately 5.2 cm on a map where one centimetre represents two hundred metres. The represented route is approximately 1,040 metres. The approximation comes from the measurement, not from a failure of the scale multiplication.
If the route is thickly drawn or curved, a slightly different tracing may give a slightly different estimate. Use the method and accuracy requested by the task. Do not claim that every final digit reflects information available from the drawing.
Rounding the final result is a separate reporting decision. The rounding guide explains why rounding intermediate pieces can differ from rounding a completed total.
For several measured route segments, retain reasonable precision when adding and convert consistently. If the question supplies exact segment lengths rather than asking for measurement, use those given lengths. The evidence source determines how much precision is justified, even when the calculator can display many more digits.
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If the learner measures the direct segment instead of the route, repair path selection. If they trace the route correctly but omit one leg, repair the route sum. If they use all the correct lengths but mix centimetres and metres, repair unit alignment.
If they measure a schematic drawing with no usable scale, repair the interpretation of the representation. If they convert correctly but label the result as travel time, repair the final quantity. Similar numerical answers can conceal very different causes.
Ask the student to point to what each number represents. “This is the map route length” and “this is the represented distance” make the reasoning inspectable. An unexplained multiplication may be right by habit without being understood.
Keep the correction specific and optimistic. A learner who knows the scale conversion has already mastered a substantial part of the task. One missing route decision does not justify treating the entire topic as insecure. Use a fresh example to check the repaired step independently.
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Knowing that two points are 1.25 kilometres apart directly does not determine the length of every possible route between them. A permitted route might follow that straight segment, or it might detour through additional points and be longer.
If no route geometry or route length is supplied, the exact journey length may not be determined. A student should not invent a convenient rectangular detour or assume a direct path simply to produce a number.
The insufficient-information guide develops a useful test: can two different routes satisfy the given information while having different lengths? If so, the exact route answer requires more information.
This is a successful reasoning conclusion, not an arithmetic failure. State what is known, such as the direct separation or a lower bound, and what remains unspecified. An honest limitation is preferable to a precise-looking journey length that depends on an unstated path assumption.
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Use one simple drawing and ask two questions: find the direct separation, then find the route through the marked corner. Keeping the diagram fixed makes the wording change visible. The student can see why two correct calculations produce different answers.
Next keep the route fixed and change only the scale. Then keep the scale fixed and introduce a required checkpoint. Each controlled variation tests a particular decision instead of changing every feature at once.
The Clementi Secondary 1 Mathematics small-group guide provides the broader teaching context. Effective support identifies the reading or calculation decision that needs attention and checks it on a new example.
Ask for a final sentence naming the path and unit. “The journey via C is 1.2 kilometres” is clearer than an isolated 1.2. The label demonstrates that the result has been returned to the problem rather than left as an unconnected calculator output.
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Suppose an invented route measures 5.5 centimetres on an accurate uniform drawing and is stated to represent 1.1 kilometres. Convert the represented length to centimetres: 1.1 kilometres is 110,000 centimetres. The matching ratio is therefore 5.5:110,000.
Dividing both sides by 5.5 gives 1:20,000. Equivalently, one centimetre represents two hundred metres. This backward calculation uses a map length and a represented length that describe the same route, with the same endpoints and path.
If the direct segment between the endpoints measures 4.4 centimetres, pairing that with the 1.1-kilometre route would falsely produce 1:25,000. The arithmetic of that ratio is correct, but the quantities do not correspond. A scale must compare matching lengths, not any two convenient numbers from the diagram.
Under the correct scale, the 4.4-centimetre direct segment represents 880 metres. The route remains 1,100 metres, giving an excess of 220 metres. The two measurements now fit one consistent drawing and one common conversion factor.
This inverse example tests the central idea from another direction. In a forward task, the student chooses a map path before converting. In a scale-finding task, the student must choose the matching map and represented paths before forming the ratio. The correspondence decision comes before either multiplication or division.
For a final check, multiply 5.5 by twenty thousand to recover 110,000 centimetres. Then convert back to 1.1 kilometres. The result confirms the ratio calculation, while the path labels confirm that the ratio was built from the right pair of lengths.
If the given map length is approximate, the recovered scale may also be approximate. Do not report an exact scale from an uncertain ruler estimate unless the task defines the values that way. Keep the distinction between a supplied exact model and an inferred measurement visible in the conclusion.
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An invented network offers two paths from A to B. The first goes A to C to D to B, with segment lengths two, three and two map centimetres. The second goes A to E to B, with lengths four and two map centimetres. Their totals are seven and six.
At a common scale where one centimetre represents two hundred metres, the routes represent 1,400 and 1,200 metres. If both paths are permitted and the question asks only for the shorter distance, the second is shorter by two hundred metres.
If the question requires passing through C, the second path is not an eligible answer even though its total is smaller. The permitted connections and checkpoints define the candidate routes. The shortest direct segment on the page cannot replace those connections unless the model allows it.
The network may also state a direction restriction, such as allowing one connection only from C to D. Treat that as part of the invented mathematical specification. Do not infer an opposite-direction connection merely because a line is visible. This is a diagram-reading exercise, not a claim about any actual road rule.
If a stated journey goes along one segment and then back along it, count both traversals. The segment exists once in the drawing, but the travelled distance includes every time it is used. A route description records movement, whereas the network picture records available connections.
For example, A to C and back to A along a two-centimetre segment contributes four map centimetres to the journey. Its final direct separation is zero, but its accumulated path is positive. This reinforces the difference between a fixed drawing element and the distance produced by following it.
No advanced route-finding algorithm is needed for these small examples. List the permitted candidate paths, add their supplied lengths and compare the requested totals. A clear route label beside each sum is enough to make the choice reviewable and prevent a prohibited shortcut from being mistaken for a valid minimum.
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Is the straight-line distance ever the journey length?
Yes, if the permitted journey follows that direct segment without detouring or backtracking in the stated flat model. The warning is that this must be established from the task, not assumed from the endpoints alone.
Can I add map lengths before converting?
Yes, when the segments are measured in matching units under the same uniform scale. Adding first and converting once gives the same exact result as converting each segment accurately and then adding. Early rounding can create a reporting difference.
What if two maps use different scales?
Convert each measured length using its own scale before comparing represented distances. A longer line on a larger-scale drawing can represent a shorter actual distance. Bare centimetre measurements from different scales are not directly comparable.
How much longer is the opening route in percentage terms?
The route is seven map centimetres compared with a direct five, so the excess is two. Relative to the direct separation, the route is 2/5 = 40% longer. The same comparison holds after scale conversion: five hundred divided by 1,250 is forty percent. If instead the question asks what share of the complete route is excess over the direct separation, the reference is seven and the fraction is 2/7, approximately 28.6%. Name the reference length before reporting the percentage. The unchanged scale factor cancels in either ratio, but the two denominator choices answer different questions.
What should I check first?
Choose the requested path, preserve its segments, and convert the resulting length with matching units. Return to the Mathematics explanation directory when a related reading or measurement decision needs another focused example.